#ifndef INCLUDE_MODE
  #define INCLUDE_MODE
  // #define REACTIVE
  // #define USE_GETLINE
  /* #define SUBMIT_ONLY */
  #define DEBUG_OUTPUT
  // #define SAMPLE_CHECK dummy
#endif
#ifdef INCLUDE_MAIN

VO Solve()
{
  CIN( int , N );
  DynamicMod::SetModulo( N );
  DynamicMod ten{ 10 };
  constexpr PrimeEnumeration<10000> pe{};
  auto div = EnumerateDivisor( pe , get<0>( EulerFunction( pe , N ) ) );
  Sort( div );
  RUN( div , d ){
    if( Power( ten , d ) == 1 ){
      RETURN( d );
    }
  }
}
REPEAT_MAIN(1);

#else /* INCLUDE_MAIN */
#ifdef INCLUDE_SUB

/* 圧縮時は中身だけ削除する。*/
IN VO Experiment()
{
}

/* 圧縮時は中身だけ削除する。*/
IN VO SmallTest()
{
}

/* 圧縮時は中身だけ削除する。*/
IN VO RandomTest( const int& test_case_num )
{
}

#define INCLUDE_MAIN
#include __FILE__
#else /* INCLUDE_SUB */
#ifdef INCLUDE_LIBRARY

/* VVV 常設でないライブラリは以下に挿入する。*/
#ifdef DEBUG
  #include "c:/Users/user/Documents/Programming/Mathematics/Arithmetic/Divisor/Enumeration/a_Body.hpp"
#else
TE <int val_limit,int le_max = val_limit>CL PrimeEnumeration{PU:bool m_is_composite[val_limit];int m_val[le_max];int m_le;CE PrimeEnumeration();IN CRI OP[](CRI i)CO;CE CRI Get(CRI i)CO;CE CO bool& IsComposite(CRI n)CO;CE CRI length()CO NE;};
TE <int val_limit,int le_max> CE PrimeEnumeration<val_limit,le_max>::PrimeEnumeration():m_is_composite(),m_val(),m_le(0){for(int i = 2;i < val_limit;i++){if(! m_is_composite[i]){if(i <=(val_limit - 1)/ i){for(int j = i * i;j < val_limit;j += i){m_is_composite[j]= true;}}m_val[m_le++]= i;if(m_le >= le_max){break;}}}}TE <int val_limit,int le_max> IN CRI PrimeEnumeration<val_limit,le_max>::OP[](CRI i)CO{AS(0 <= i && i < m_le);RE m_val[i];}TE <int val_limit,int le_max> CE CRI PrimeEnumeration<val_limit,le_max>::Get(CRI i)CO{RE m_val[i];}TE <int val_limit,int le_max> CE CO bool& PrimeEnumeration<val_limit,le_max>::IsComposite(CRI n)CO{RE m_is_composite[n];}TE <int val_limit,int le_max> CE CRI PrimeEnumeration<val_limit,le_max>::length()CO NE{RE m_le;}
CL HeapPrimeEnumeration{PU:int m_val_limit;VE<bool> m_is_composite;VE<int> m_val;int m_le;IN HeapPrimeEnumeration(CRI val_limit);IN CRI OP[](CRI i)CO;IN CRI Get(CRI i)CO;IN bool IsComposite(CRI n)CO;IN CRI length()CO NE;};
IN HeapPrimeEnumeration::HeapPrimeEnumeration(CRI val_limit):m_val_limit(val_limit),m_is_composite(m_val_limit),m_val(),m_le(0){for(int i = 2;i < m_val_limit;i++){if(! m_is_composite[i]){if(i <=(m_val_limit - 1)/ i){for(int j = i * i;j < val_limit;j += i){m_is_composite[j]= true;}}m_val.push_back(i);}}m_le = m_val.SZ();}IN CRI HeapPrimeEnumeration::OP[](CRI i)CO{AS(0 <= i && i < m_le);RE m_val[i];}IN CRI HeapPrimeEnumeration::Get(CRI i)CO{RE OP[](i);}IN bool HeapPrimeEnumeration::IsComposite(CRI n)CO{AS(0 <= n && n < m_val_limit);RE m_is_composite[n];}IN CRI HeapPrimeEnumeration::length()CO NE{RE m_le;}

TE <TY PE> auto CheckPE(CO PE& pe)-> decltype(pe.IsComposite(0),true_type());TE <TY...> false_type CheckPE(...);TE <TY T>CE bool IsPE = decltype(CheckPE(declval<T>()))();

TE <int val_limit>CL LeastDivisor{PU:int m_val[val_limit];CE LeastDivisor()NE;IN CRI OP[](CRI i)CO;CE CRI Get(CRI i)CO;CE int length()CO NE;};
TE <int val_limit> CE LeastDivisor<val_limit>::LeastDivisor()NE:m_val{}{for(int d = 2;d < val_limit;d++){if(m_val[d]== 0){for(int n = d;n < val_limit;n += d){m_val[n]== 0?m_val[n]= d:d;}}}}TE <int val_limit> IN CRI LeastDivisor<val_limit>::OP[](CRI i)CO{AS(0 <= i && i < val_limit);RE m_val[i];}TE <int val_limit> CE CRI LeastDivisor<val_limit>::Get(CRI i)CO{RE m_val[i];}TE <int val_limit> CE int LeastDivisor<val_limit>::length()CO NE{RE val_limit;}
CL HeapLeastDivisor{PU:int m_val_limit;VE<int> m_val;IN HeapLeastDivisor(CRI val_limit)NE;IN CRI OP[](CRI i)CO;IN CRI Get(CRI i)CO;IN CRI length()CO NE;};
IN HeapLeastDivisor::HeapLeastDivisor(CRI val_limit)NE:m_val_limit(val_limit),m_val(m_val_limit){for(int d = 2;d < m_val_limit;d++){if(m_val[d]== 0){for(int n = d;n < m_val_limit;n += d){m_val[n]== 0?m_val[n]= d:d;}}}}IN CRI HeapLeastDivisor::OP[](CRI i)CO{AS(0 <= i && i < m_val_limit);RE m_val[i];}IN CRI HeapLeastDivisor::Get(CRI i)CO{RE m_val[i];}IN CRI HeapLeastDivisor::length()CO NE{RE m_val_limit;}

TE <TY PE,TY INT>auto PrimeFactorisation(CO PE& pe,INT n)-> enable_if_t<IsPE<PE>,pair<VE<INT>,VE<int>>>{AS(n > 0);VE<INT> P{};VE<int> E{};CRI le = pe.length();for(int i = 0;i < le;i++){auto& p = pe[i];if(n % p == 0){int e = 1;WH((n /= p)% p == 0){e++;}P.push_back(p);E.push_back(e);}else if(n / p < p){break;}}if(n != 1){P.push_back(n);E.push_back(1);}RE{MO(P),MO(E)};}TE <TY LD>auto PrimeFactorisation(CO LD& ld,int n)-> enable_if_t<!IsPE<LD>,pair<VE<int>,VE<int>>>{AS(n > 0);VE<int> P{};VE<int> E{};if(n > 1){P.push_back(ld[n]);E.push_back(1);n /= ld[n];}WH(n > 1){if(P.back()!= ld[n]){P.push_back(ld[n]);E.push_back(1);}else{E.back()++;}n /= ld[n];}RE{MO(P),MO(E)};}TE <TY PE,TY INT>auto PrimePowerFactorisation(CO PE& pe,INT n)-> enable_if_t<IsPE<PE>,tuple<VE<INT>,VE<int>,VE<INT>>>{AS(n > 0);VE<INT> P{};VE<int> E{};VE<INT> Q{};CRI le = pe.length();for(int i = 0;i < le;i++){auto& p = pe[i];if(n % p == 0){int e = 1;INT q = p;WH((n /= p)% p == 0){e++;q *= p;}P.push_back(p);E.push_back(e);Q.push_back(q);}else if(n / p < p){break;}}if(n != 1){P.push_back(n);E.push_back(1);Q.push_back(n);}RE{MO(P),MO(E),MO(Q)};}TE <TY LD>auto PrimePowerFactorisation(CO LD& ld,int n)-> enable_if_t<!IsPE<LD>,tuple<VE<int>,VE<int>,VE<int>>>{AS(n > 0);VE<int> P{};VE<int> E{};VE<int> Q{};if(n > 1){P.push_back(ld[n]);E.push_back(1);Q.push_back(ld[n]);n /= ld[n];}WH(n > 1){if(P.back()!= ld[n]){P.push_back(ld[n]);E.push_back(1);Q.push_back(ld[n]);}else{Q.back()*= ld[n];E.back()++;}n /= ld[n];}RE{MO(P),MO(E),MO(Q)};}

TE <TY INT> INT CountDivisorBody(VE<int>& E)NE{CO int LE = E.SZ();INT AN = 1;for(int i = 0;i < LE;i++){AN *= ++E[i];}RE AN;}TE <TY PE,TY INT>INT CountDivisor(CO PE& pe,INT n)NE{auto[P,E]= PrimeFactorisation(pe,MO(n));RE CountDivisorBody<INT>(E);}

TE <TY INT> VE<INT> EnumerateDivisorBody(CO VE<INT>& P,VE<int>& E){CO int le = P.SZ();VE AN(CountDivisorBody<INT>(E),INT(1));int SZ = 1;for(int i = 0;i < le;i++){auto& P_i = P[i];auto& E_i = E[i];INT q = 1;int j_shift = 0;for(int e = 1;e < E_i;e++){q *= P_i;j_shift += SZ;for(int j = 0;j < SZ;j++){AN[j + j_shift]= AN[j]* q;}}SZ *= E_i;}RE AN;}TE <TY PE,TY INT>auto EnumerateDivisor(CO PE& pe,INT n)-> enable_if_t<IsPE<PE>,VE<INT>>{auto[P,E]= PrimeFactorisation(pe,MO(n));RE EnumerateDivisorBody(P,E);}TE <TY LD,TY INT>auto EnumerateDivisor(CO LD& ld,INT n)-> enable_if_t<!IsPE<LD>,VE<INT>>{VE<INT> P{};VE<int> E{};WH(n > 1){auto& p = ld[n];int e = 1;WH((n /= p)% p == 0){e++;}P.push_back(p);E.push_back(e);}RE EnumerateDivisorBody(P,E);}TE <TY INT>VE<VE<INT>> TotalEnumerateDivisor(CO INT& SZ)NE{VE<VE<INT>> AN(SZ);for(INT d = 1;d < SZ;d++){for(INT n = 0;n < SZ;n += d){AN[n].push_back(d);}}RE AN;}TE <TY INT,TY FUNC,TY U>VE<U> TotalEnumerateDivisor(CO INT& SZ,FUNC f,CO U& init)NE{ST_AS(is_invocable_r_v<U,FUNC,U,CO INT&>);VE<U> AN(SZ,init);for(INT d = 1;d < SZ;d++){for(INT n = 0;n < SZ;n += d){AN[n]= f(MO(AN[n]),d);}}RE AN;}
#endif

#ifdef DEBUG
  #include "c:/Users/user/Documents/Programming/Mathematics/Arithmetic/Mod/Function/Euler/a_Body.hpp"
#else
TE <TY PF,TY INT>tuple<INT,VE<INT>,VE<int>> EulerFunction_Body(PF pf,CO INT& n){auto[P,E]= pf(n);INT AN = n;for(auto& p:P){AN -= AN / p;}RE{AN,MO(P),MO(E)};}TE <TY PE,TY INT> IN tuple<INT,VE<int>,VE<int>> EulerFunction(CO PE& pe,CO INT& n){RE EulerFunction_Body([&](CRI i){RE PrimeFactorisation(pe,i);},n);}TE <TY PE,TY INT>VE<INT> TotalEulerFunction(CO PE& pe,CO INT& n_max){VE<INT> AN(n_max + 1);for(INT n = 1;n <= n_max;n++){AN[n]= n;}auto quotient = AN;CRI le = pe.le();for(int i = 0;i < le;i++){auto& p_i = pe[i];INT n = 0;WH((n += p_i)<= n_max){INT& AN_n = AN[n];INT& quotient_n = quotient[n];AN_n -= AN_n / p_i;WH((quotient_n /= p_i)% p_i == 0){}}}for(INT n = le == 0?2:pe[le - 1];n <= n_max;n++){CO INT& quotient_n = quotient[n];if(quotient_n != 1){INT& AN_n = AN[n];AN_n -= AN_n / quotient_n;}}RE AN;}
#endif

#ifdef DEBUG
  #include "c:/Users/user/Documents/Programming/Mathematics/Arithmetic/Mod/DynamicModulo/Debug/a_Body.hpp"
#else
TE <TY INT1,TY INT2> CE INT1 Residue(INT1 n,CO INT2& M)NE{RE MO(n < 0?((((++n)*= -1)%= M)*= -1)+= M - 1:n < M?n:n %= M);}

TE <int NUM> CL DynamicMods;TE <int NUM>CL COantsForDynamicMods{PU:COantsForDynamicMods()= delete;ST uint g_M;ST uint g_M_minus;ST int g_order;ST int g_order_minus;ST bool g_M_is_prime;};
TE <int NUM> uint COantsForDynamicMods<NUM>::g_M = 0;TE <int NUM> uint COantsForDynamicMods<NUM>::g_M_minus = -1;TE <int NUM> int COantsForDynamicMods<NUM>::g_order = 1;TE <int NUM> int COantsForDynamicMods<NUM>::g_order_minus = 0;TE <int NUM> bool COantsForDynamicMods<NUM>::g_M_is_prime = false;

#define DC_OF_CM_FOR_DYNAMIC_MOD(OPR)IN bool OP OPR(CO DynamicMods<NUM>& n)CO NE
#define DC_OF_AR_FOR_DYNAMIC_MOD(OPR,EX)IN DynamicMods<NUM> OP OPR(DynamicMods<NUM> n)CO EX;
#define DF_OF_CM_FOR_DYNAMIC_MOD(OPR)TE <int NUM> IN bool DynamicMods<NUM>::OP OPR(CO DynamicMods<NUM>& n)CO NE{RE m_n OPR n.m_n;}
#define DF_OF_AR_FOR_DYNAMIC_MOD(OPR,EX,LEFT,OPR2)TE <int NUM> IN DynamicMods<NUM> DynamicMods<NUM>::OP OPR(DynamicMods<NUM> n)CO EX{RE MO(LEFT OPR2 ## = *TH);}TE <int NUM,TY T> IN DynamicMods<NUM> OP OPR(T n0,CO DynamicMods<NUM>& n1)EX{RE MO(DynamicMods<NUM>(MO(n0))OPR ## = n1);}
TE <int NUM>CL DynamicMods{PU:uint m_n;IN DynamicMods()NE;IN DynamicMods(CO DynamicMods<NUM>& n)NE;IN DynamicMods(DynamicMods<NUM>&& n)NE;TE <TY T> IN DynamicMods(T n)NE;IN DynamicMods<NUM>& OP=(DynamicMods<NUM> n)NE;IN DynamicMods<NUM>& OP+=(CO DynamicMods<NUM>& n)NE;IN DynamicMods<NUM>& OP-=(CO DynamicMods<NUM>& n)NE;IN DynamicMods<NUM>& OP*=(CO DynamicMods<NUM>& n)NE;IN DynamicMods<NUM>& OP/=(DynamicMods<NUM> n);TE <TY INT> IN DynamicMods<NUM>& OP<<=(INT n);TE <TY INT> IN DynamicMods<NUM>& OP>>=(INT n);IN DynamicMods<NUM>& OP++()NE;IN DynamicMods<NUM> OP++(int)NE;IN DynamicMods<NUM>& OP--()NE;IN DynamicMods<NUM> OP--(int)NE;DC_OF_CM_FOR_DYNAMIC_MOD(==);DC_OF_CM_FOR_DYNAMIC_MOD(!=);DC_OF_CM_FOR_DYNAMIC_MOD(<);DC_OF_CM_FOR_DYNAMIC_MOD(<=);DC_OF_CM_FOR_DYNAMIC_MOD(>);DC_OF_CM_FOR_DYNAMIC_MOD(>=);DC_OF_AR_FOR_DYNAMIC_MOD(+,NE);DC_OF_AR_FOR_DYNAMIC_MOD(-,NE);DC_OF_AR_FOR_DYNAMIC_MOD(*,NE);DC_OF_AR_FOR_DYNAMIC_MOD(/,);TE <TY INT> IN DynamicMods<NUM> OP^(INT EX)CO;TE <TY INT> IN DynamicMods<NUM> OP<<(INT n)CO;TE <TY INT> IN DynamicMods<NUM> OP>>(INT n)CO;IN DynamicMods<NUM> OP-()CO NE;IN DynamicMods<NUM>& SignInvert()NE;IN DynamicMods<NUM>& Invert();TE <TY INT> IN DynamicMods<NUM>& PW(INT EX);IN VO swap(DynamicMods<NUM>& n)NE;IN CRUI RP()CO NE;ST IN DynamicMods<NUM> DeRP(uint n)NE;ST IN CO DynamicMods<NUM>& Inverse(CRUI n);ST IN CO DynamicMods<NUM>& Factorial(CRUI n);ST IN CO DynamicMods<NUM>& FactorialInverse(CRUI n);ST IN DynamicMods<NUM> Combination(CRUI n,CRUI i);ST IN CO DynamicMods<NUM>& zero()NE;ST IN CO DynamicMods<NUM>& one()NE;ST IN CRUI GetModulo()NE;ST IN VO SetModulo(CRUI M,CRI order_minus = -1)NE;TE <TY INT> IN DynamicMods<NUM>& PositivePW(INT EX)NE;TE <TY INT> IN DynamicMods<NUM>& NonNegativePW(INT EX)NE;US COants = COantsForDynamicMods<NUM>;};
US DynamicMod = DynamicMods<0>;
TE <int NUM> IN DynamicMods<NUM>::DynamicMods()NE:m_n(){}TE <int NUM> IN DynamicMods<NUM>::DynamicMods(CO DynamicMods<NUM>& n)NE:m_n(n.m_n){}TE <int NUM> IN DynamicMods<NUM>::DynamicMods(DynamicMods<NUM>&& n)NE:m_n(MO(n.m_n)){}TE <int NUM> TE <TY T> IN DynamicMods<NUM>::DynamicMods(T n)NE:m_n(Residue(uint(MO(n)),COants::g_M)){ST_AS(is_COructible_v<uint,decay_t<T> >);}TE <int NUM> IN DynamicMods<NUM>& DynamicMods<NUM>::OP=(DynamicMods<NUM> n)NE{m_n = MO(n.m_n);RE *TH;}TE <int NUM> IN DynamicMods<NUM>& DynamicMods<NUM>::OP+=(CO DynamicMods<NUM>& n)NE{(m_n += n.m_n)< COants::g_M?m_n:m_n -= COants::g_M;RE *TH;}TE <int NUM> IN DynamicMods<NUM>& DynamicMods<NUM>::OP-=(CO DynamicMods<NUM>& n)NE{m_n < n.m_n?(m_n += COants::g_M)-= n.m_n:m_n -= n.m_n;RE *TH;}TE <int NUM> IN DynamicMods<NUM>& DynamicMods<NUM>::OP*=(CO DynamicMods<NUM>& n)NE{m_n = Residue(MO(ull(m_n)* n.m_n),COants::g_M);RE *TH;}TE <int NUM> IN DynamicMods<NUM>& DynamicMods<NUM>::OP/=(DynamicMods<NUM> n){RE OP*=(n.Invert());}TE <int NUM> TE <TY INT> IN DynamicMods<NUM>& DynamicMods<NUM>::OP<<=(INT n){AS(n >= 0);RE *TH *= DynamicMods<NUM>(2).NonNegativePW(MO(n));}TE <int NUM> TE <TY INT> IN DynamicMods<NUM>& DynamicMods<NUM>::OP>>=(INT n){AS(n >= 0);WH(n-- > 0){((m_n & 1)== 0?m_n:m_n += COants::g_M)>>= 1;}RE *TH;}TE <int NUM> IN DynamicMods<NUM>& DynamicMods<NUM>::OP++()NE{m_n < COants::g_M_minus?++m_n:m_n = 0;RE *TH;}TE <int NUM> IN DynamicMods<NUM> DynamicMods<NUM>::OP++(int)NE{DynamicMods<NUM> n{*TH};OP++();RE n;}TE <int NUM> IN DynamicMods<NUM>& DynamicMods<NUM>::OP--()NE{m_n == 0?m_n = COants::g_M_minus:--m_n;RE *TH;}TE <int NUM> IN DynamicMods<NUM> DynamicMods<NUM>::OP--(int)NE{DynamicMods<NUM> n{*TH};OP--();RE n;}DF_OF_CM_FOR_DYNAMIC_MOD(==);DF_OF_CM_FOR_DYNAMIC_MOD(!=);DF_OF_CM_FOR_DYNAMIC_MOD(>);DF_OF_CM_FOR_DYNAMIC_MOD(>=);DF_OF_CM_FOR_DYNAMIC_MOD(<);DF_OF_CM_FOR_DYNAMIC_MOD(<=);DF_OF_AR_FOR_DYNAMIC_MOD(+,NE,n,+);DF_OF_AR_FOR_DYNAMIC_MOD(-,NE,n.SignInvert(),+);DF_OF_AR_FOR_DYNAMIC_MOD(*,NE,n,*);DF_OF_AR_FOR_DYNAMIC_MOD(/,,n.Invert(),*);TE <int NUM> TE <TY INT> IN DynamicMods<NUM> DynamicMods<NUM>::OP^(INT EX)CO{RE MO(DynamicMods<NUM>(*TH).PW(MO(EX)));}TE <int NUM> TE <TY INT> IN DynamicMods<NUM> DynamicMods<NUM>::OP<<(INT n)CO{RE MO(DynamicMods<NUM>(*TH)<<= MO(n));}TE <int NUM> TE <TY INT> IN DynamicMods<NUM> DynamicMods<NUM>::OP>>(INT n)CO{RE MO(DynamicMods<NUM>(*TH)>>= MO(n));}TE <int NUM> IN DynamicMods<NUM> DynamicMods<NUM>::OP-()CO NE{RE MO(DynamicMods<NUM>(*TH).SignInvert());}TE <int NUM> IN DynamicMods<NUM>& DynamicMods<NUM>::SignInvert()NE{m_n > 0?m_n = COants::g_M - m_n:m_n;RE *TH;}TE <int NUM> IN DynamicMods<NUM>& DynamicMods<NUM>::Invert(){RE m_n <(COants::g_M_is_prime?1e6:3e4)?*TH = Inverse(m_n):NonNegativePW(COants::g_order_minus);}TE <int NUM> TE <TY INT> IN DynamicMods<NUM>& DynamicMods<NUM>::PositivePW(INT EX)NE{DynamicMods<NUM> PW{*TH};EX--;WH(EX != 0){(EX & 1)== 1?*TH *= PW:*TH;EX >>= 1;PW *= PW;}RE *TH;}TE <int NUM> TE <TY INT> IN DynamicMods<NUM>& DynamicMods<NUM>::NonNegativePW(INT EX)NE{RE EX == 0?(m_n = 1,*TH):PositivePW(MO(EX));}TE <int NUM> TE <TY INT> IN DynamicMods<NUM>& DynamicMods<NUM>::PW(INT EX){bool neg = EX < 0;AS(!(neg && m_n == 0));RE NonNegativePW(MO(neg?(EX %= COants::g_order)== 0?EX:EX += COants::g_order:EX));}TE <int NUM> IN VO DynamicMods<NUM>::swap(DynamicMods<NUM>& n)NE{std::swap(m_n,n.m_n);}TE <int NUM> IN CO DynamicMods<NUM>& DynamicMods<NUM>::Inverse(CRUI n){ST VE<DynamicMods<NUM>> memory ={zero(),one()};ST uint LE_curr = 2;AS(COants::g_M == 1||(0 < n && n < COants::g_M));WH(LE_curr <= n){memory.push_back(COants::g_M_is_prime?DeRP(COants::g_M - memory[COants::g_M % LE_curr].m_n * ull(COants::g_M / LE_curr)% COants::g_M):DeRP(n).NonNegativePW(COants::g_order_minus));LE_curr++;}RE memory[n];}TE <int NUM> IN CO DynamicMods<NUM>& DynamicMods<NUM>::Factorial(CRUI n){ST VE<DynamicMods<NUM>> memory ={one(),one()};ST uint LE_curr = 2;if(COants::g_M <= n){RE zero();}WH(LE_curr <= n && memory.back().m_n != 0){memory.push_back(memory.back()* DeRP(LE_curr));LE_curr++;}RE LE_curr <= n?memory.back():memory[n];}TE <int NUM> IN CO DynamicMods<NUM>& DynamicMods<NUM>::FactorialInverse(CRUI n){ST VE<DynamicMods<NUM>> memory ={one(),one()};ST uint LE_curr = 2;WH(LE_curr <= n){memory.push_back(memory[LE_curr - 1]* Inverse(LE_curr));LE_curr++;}RE memory[n];}TE <int NUM> IN DynamicMods<NUM> DynamicMods<NUM>::Combination(CRUI n,CRUI i){RE i <= n?Factorial(n)* FactorialInverse(i)* FactorialInverse(n - i):zero();}TE <int NUM> IN CRUI DynamicMods<NUM>::RP()CO NE{RE m_n;}TE <int NUM> IN DynamicMods<NUM> DynamicMods<NUM>::DeRP(uint n)NE{DynamicMods<NUM> n_copy{};n_copy.m_n = MO(n);RE n_copy;}TE <int NUM> IN CO DynamicMods<NUM>& DynamicMods<NUM>::zero()NE{ST CO DynamicMods<NUM> z{};RE z;}TE <int NUM> IN CO DynamicMods<NUM>& DynamicMods<NUM>::one()NE{ST CO DynamicMods<NUM> o{1};RE o;}TE <int NUM> IN CRUI DynamicMods<NUM>::GetModulo()NE{RE COants::g_M;}TE <int NUM> IN VO DynamicMods<NUM>::SetModulo(CRUI M,CRI order_minus)NE{COants::g_M = M;COants::g_M_minus = M - 1;COants::g_order = order_minus == -1?M - 1:order_minus+1;COants::g_order_minus = COants::g_order-1;COants::g_M_is_prime = order_minus == -1;}TE <int NUM> IN DynamicMods<NUM> Inverse(CO DynamicMods<NUM>& n){RE MO(DynamicMods<NUM>(n).Invert());}TE <int NUM,TY INT> IN DynamicMods<NUM> PW(DynamicMods<NUM> n,INT EX){RE MO(n.PW(MO(EX)));}TE <int NUM> IN VO swap(DynamicMods<NUM>& n0,DynamicMods<NUM>& n1)NE{n0.swap(n1);}TE <int NUM> IN string to_string(CO DynamicMods<NUM>& n)NE{RE to_string(n.RP())+ " + " + to_string(DynamicMods<NUM>::GetModulo())+ "Z";}TE <int NUM,CL Traits> IN IS& OP>>(IS& is,DynamicMods<NUM>& n){ll m;is >> m;n = m;RE is;}TE <int NUM,CL Traits> IN OS& OP<<(OS& os,CO DynamicMods<NUM>& n){RE os << n.RP();}

TE <TY INT1,TY INT2>INT1 GCD(CO INT1& b_0,CO INT2& b_1){INT1 a_0 = b_0 < 0?-b_0:b_0;INT1 a_1 = b_1 < 0?-b_1:b_1;WH(a_1 != 0){swap(a_0 %= a_1,a_1);}RE a_0;}TE <TY INT1,TY INT2> IN INT1 LCM(CO INT1& b_0,CO INT2& b_1){RE(b_0 == 0 && b_1 == 0)?0:(b_0 / GCD(b_0,b_1))* b_1;}
#endif

/* AAA 常設でないライブラリは以上に挿入する。*/

#define INCLUDE_SUB
#include __FILE__
#else /* INCLUDE_LIBRARY */
#ifdef DEBUG
  #define _GLIBCXX_DEBUG
#else
  #pragma GCC optimize ( "O3" )
  #pragma GCC optimize ( "unroll-loops" )
  #pragma GCC target ( "sse4.2,fma,avx2,popcnt,lzcnt,bmi2" )
  #define REPEAT_MAIN( BOUND ) START_MAIN; CEXPR( int , bound_test_case_num , BOUND ); int test_case_num = 1; if CE( bound_test_case_num > 1 ){ SET_ASSERT( test_case_num , 1 , bound_test_case_num ); } FINISH_MAIN
  #define FINISH_MAIN REPEAT( test_case_num ){ if CE( bound_test_case_num > 1 ){ CERR( "testcase " , VARIABLE_FOR_REPEAT_test_case_num , ":" ); } Solve(); CERR( "" ); } }
  #define DEXPR( LL , BOUND , VALUE1 , VALUE2 ) CEXPR( LL , BOUND , VALUE1 )
  #define ASSERT( A , MIN , MAX ) AS( ( MIN ) <= A && A <= ( MAX ) )
  #ifdef USE_GETLINE
    #define SET_SEPARATE( SEPARATOR , ... ) VariadicGetline( cin , SEPARATOR , __VA_ARGS__ )
    #define SET( ... ) SET_SEPARATE( '\n' , __VA_ARGS__ )
    #define GETLINE_SEPARATE( SEPARATOR , ... ) string __VA_ARGS__; SET_SEPARATE( SEPARATOR , __VA_ARGS__ )
    #define GETLINE( ... ) GETLINE_SEPARATE( '\n' , __VA_ARGS__ )
  #else
    #define SET( ... ) VariadicCin( cin , __VA_ARGS__ )
    #define CIN( LL , ... ) LL __VA_ARGS__; SET( __VA_ARGS__ )
    #define SET_A( I , N , ... ) VariadicResize( N + I , __VA_ARGS__ ); FOR( VARIABLE_FOR_SET_A , 0 , N ){ VariadicSet( cin , VARIABLE_FOR_SET_A + I , __VA_ARGS__ ); }
    #define CIN_A( LL , I , N , ... ) VE<LL> __VA_ARGS__; SET_A( I , N , __VA_ARGS__ )
    #define CIN_AA( LL , I0 , N0 , I1 , N1 , VAR ) VE<VE<LL>> VAR( N0 + I0 ); FOR( VARIABLE_FOR_CIN_AA , 0 , N0 ){ SET_A( I1 , N1 , VAR[VARIABLE_FOR_CIN_AA + I0] ); }
  #endif
  #define SET_ASSERT( A , MIN , MAX ) SET( A ); ASSERT( A , MIN , MAX )
  #define SOLVE_ONLY 
  #define COUT( ... ) VariadicCout( cout , __VA_ARGS__ ) << ENDL
  #define COUTNS( ... ) VariadicCoutNonSep( cout , __VA_ARGS__ )
  #define CERR( ... ) 
  #define CERRNS( ... ) 
  #define COUT_A( I , N , A ) CoutArray( cout , I , N , A ) << ENDL
  #define CERR_A( I , N , A ) 
  #define TLE( CONDITION ) if( !( CONDITION ) ){ ll TLE_VAR = 1; while( TLE_VAR != 0 ){ ( TLE_VAR += 2 ) %= int( 1e9 ); } COUT( TLE_VAR ); }
  #define MLE( CONDITION ) if( !( CONDITION ) ){ vector<vector<ll>> MLE_VAR{}; REPEAT( 1e6 ){ MLE_VAR.push_back( vector<ll>( 1e6 ) ); } COUT( MLE_VAR ); }
  #define OLE( CONDITION ) if( !( CONDITION ) ){ REPEAT( 1e8 ){ COUT( "OLE" ); } }
#endif
#ifdef REACTIVE
  #ifdef DEBUG
    #define RSET( A , ... ) A = __VA_ARGS__
  #else
    #define RSET( A , ... ) SET( A )
  #endif
  #define RCIN( LL , A , ... ) LL A; RSET( A , __VA_ARGS__ )
  #define ENDL endl
#else
  #define ENDL "\n"
#endif
#include <bits/stdc++.h>
using namespace std;
#define ATT __attribute__( ( target( "sse4.2,fma,avx2,popcnt,lzcnt,bmi2" ) ) )
#define START_MAIN int main(){ ios_base::sync_with_stdio( false ); cin.tie( nullptr )
#define START_WATCH chrono::system_clock::time_point watch = chrono::system_clock::now(); double loop_average_time = 0.0 , loop_start_time = 0.0 , current_time = 0.0; int loop_count = 0
#define CURRENT_TIME ( current_time = static_cast<double>( chrono::duration_cast<chrono::microseconds>( chrono::system_clock::now() - watch ).count() / 1000.0 ) )
#define CHECK_WATCH( TL_MS ) ( CURRENT_TIME , loop_count == 0 ? loop_start_time = current_time : loop_average_time = ( current_time - loop_start_time ) / loop_count , ++loop_count , current_time < TL_MS - loop_average_time * 2 - 100.0 )
#define CEXPR( LL , BOUND , VALUE ) CE LL BOUND = VALUE
#define SET_A_ASSERT( I , N , A , MIN , MAX ) FOR( VARIABLE_FOR_SET_A , 0 , N ){ SET_ASSERT( A[VARIABLE_FOR_SET_A + I] , MIN , MAX ); }
#define SET_AA_ASSERT( I0 , N0 , I1 , N1 , A , MIN , MAX ) FOR( VARIABLE_FOR_SET_AA0 , 0 , N0 ){ FOR( VARIABLE_FOR_SET_AA1 , 0 , N1 ){ SET_ASSERT( A[VARIABLE_FOR_SET_AA0 + I0][VARIABLE_FOR_SET_AA1 + I1] , MIN , MAX ); } }
#define CIN_ASSERT( A , MIN , MAX ) decldecay_t( MAX ) A; SET_ASSERT( A , MIN , MAX )
#define CIN_A_ASSERT( I , N , A , MIN , MAX ) vector<decldecay_t( MAX )> A( N + I ); SET_A_ASSERT( I , N , A , MIN , MAX )
#define CIN_AA_ASSERT( I0 , N0 , I1 , N1 , A , MIN , MAX ) vector A( N0 + I0 , vector<decldecay_t( MAX )>( N1 + I1 ) ); SET_AA_ASSERT( I0 , N0 , I1 , N1 , A , MIN , MAX )
#define SET_MAX( A , X ) A = max( A , decltype( A )( X ) )
#define SET_MIN( A , X ) A = min( A , decltype( A )( X ) )
#define FOR( VAR , INITIAL , FINAL_PLUS_ONE ) for( decldecay_t( FINAL_PLUS_ONE ) VAR = INITIAL ; VAR < FINAL_PLUS_ONE ; VAR ++ )
#define FOREQ( VAR , INITIAL , FINAL ) for( decldecay_t( FINAL ) VAR = INITIAL ; VAR <= FINAL ; VAR ++ )
#define FOREQINV( VAR , INITIAL , FINAL ) for( decldecay_t( INITIAL ) VAR = INITIAL ; VAR + 1 > FINAL ; VAR -- )
#define ITR( ARRAY ) auto begin_ ## ARRAY = ARRAY .BE() , itr_ ## ARRAY = begin_ ## ARRAY , end_ ## ARRAY = ARRAY .EN()
#define FOR_ITR( ARRAY ) for( ITR( ARRAY ) , itr = itr_ ## ARRAY ; itr_ ## ARRAY != end_ ## ARRAY ; itr_ ## ARRAY ++ , itr++ )
#define RUN( ARRAY , ... ) for( auto&& __VA_ARGS__ : ARRAY )
#define REPEAT( HOW_MANY_TIMES ) FOR( VARIABLE_FOR_REPEAT , 0 , HOW_MANY_TIMES )
#define SET_PRECISION( DECIMAL_DIGITS ) cout << fixed << setprecision( DECIMAL_DIGITS ); cerr << fixed << setprecision( DECIMAL_DIGITS )
#define RETURN( ... ) SOLVE_ONLY; COUT( __VA_ARGS__ ); RE
#define COMPARE( ... ) auto naive = Naive( __VA_ARGS__ , false ); auto answer = Answer( __VA_ARGS__ , false ); bool match = naive == answer; CERR( "(" , #__VA_ARGS__ , ") == (" , __VA_ARGS__ , ") : Naive == " , naive , match ? "==" : "!=" , answer , "== Answer" ); if( !match ){ CERR( "出力の不一致が検出されました。" ); RE; }

/* 圧縮用 */
#define TE template
#define TY typename
#define US using
#define ST static
#define AS assert
#define IN inline
#define CL class
#define PU public
#define OP operator
#define CE constexpr
#define CO const
#define NE noexcept
#define RE return 
#define WH while
#define VO void
#define VE vector
#define LI list
#define BE begin
#define EN end
#define SZ size
#define LE length
#define PW Power
#define MO move
#define TH this
#define CRI CO int&
#define CRUI CO uint&
#define CRL CO ll&
#define VI virtual 
#define IS basic_istream<char,Traits>
#define OS basic_ostream<char,Traits>
#define ST_AS static_assert
#define reMO_CO remove_const
#define is_COructible_v is_constructible_v
#define rBE rbegin

/* 型のエイリアス */
#define decldecay_t(VAR)decay_t<decltype(VAR)>
TE <TY F,TY...Args> US ret_t = decltype(declval<F>()(declval<Args>()...));
TE <TY T> US inner_t = TY T::type;
US uint = unsigned int;
US ll = long long;
US ull = unsigned long long;
US ld = long double;
US lld = __float128;
TE <TY INT> US T2 = pair<INT,INT>;
TE <TY INT> US T3 = tuple<INT,INT,INT>;
TE <TY INT> US T4 = tuple<INT,INT,INT,INT>;
US path = pair<int,ll>;

/* VVV 常設ライブラリは以下に挿入する。*/
#ifdef DEBUG
  #include "C:/Users/user/Documents/Programming/Contest/Template/Local/a_Body.hpp"
#else
/* Random (1KB)*/
ll GetRand(CRI Rand_min,CRI Rand_max){AS(Rand_min <= Rand_max);ll AN = time(NULL);RE AN * rand()%(Rand_max + 1 - Rand_min)+ Rand_min;}

/* Set (2KB)*/
#ifdef DEBUG
  #include "c:/Users/user/Documents/Programming/Mathematics/Mathematics/Utility/Set/a_Body.hpp"
#else
#define DC_OF_HASH(...)struct hash<__VA_ARGS__>{IN size_t OP()(CO __VA_ARGS__& n)CO;};
CL is_ordered{PU:is_ordered()= delete;TE <TY T> ST CE auto Check(CO T& t)-> decltype(t < t,true_type());ST CE false_type Check(...);TE <TY T> ST CE CO bool value = is_same_v< decltype(Check(declval<T>())),true_type >;};
TE <TY T>US Set = conditional_t<is_COructible_v<unordered_set<T>>,unordered_set<T>,conditional_t<is_ordered::value<T>,set<T>,VO>>;

#define DF_OF_POP_FOR_SET(SET)TE <TY T> IN T pop_max(SET& S){AS(!S.empty());auto IT = --S.EN();CO T AN = MO(*IT);S.erase(IT);RE AN;}TE <TY T> IN T pop_min(SET& S){AS(!S.empty());auto IT = S.BE();CO T AN = MO(*IT);S.erase(IT);RE AN;}TE <TY T> IN SET& OP+=(SET& S,T t){S.insert(MO(t));RE S;}TE <TY T> IN SET& OP-=(SET& S,CO T& t){S.erase(t);RE S;}TE <TY T> IN CO T& Get(CO SET& S,int i){auto BE = S.BE(),EN = S.EN();auto& IT = i < 0?(++i,--EN):BE;WH(i > 0 && IT != EN){--i;++IT;}WH(i < 0 && IT != BE){++i;--IT;}AS(i == 0);RE *IT;}
#define DF_OF_UNION_FOR_SET(SET)TE <TY T> IN SET& OP|=(SET& a0,CO SET& a1){for(auto& t:a1){a0 += t;}RE a0;}TE <TY T> IN SET OP|(SET a0,CO SET& a1){RE MO(a0 |= a1);}TE <TY T> IN int len(CO SET& S){RE S.SZ();}
TE <TY SET,TY T> IN TY SET::const_iterator MaximumLeq(CO SET& S,CO T& t){auto IT = S.upper_bound(t);RE IT == S.BE()?S.EN():--IT;}TE <TY SET,TY T> IN TY SET::const_iterator MaximumLt(CO SET& S,CO T& t){auto IT = S.lower_bound(t);RE IT == S.BE()?S.EN():--IT;}TE <TY SET,TY T> IN TY SET::const_iterator MinimumGeq(CO SET& S,CO T& t){RE S.lower_bound(t);}TE <TY SET,TY T> IN TY SET::const_iterator MinimumGt(CO SET& S,CO T& t){RE S.upper_bound(t);}TE <TY SET,TY ITERATOR> IN VO EraseBack(SET& S,ITERATOR& IT){IT = S.erase(IT);}TE <TY SET,TY ITERATOR> IN VO EraseFront(SET& S,ITERATOR& IT){IT = S.erase(IT);IT == S.BE()?IT = S.EN():--IT;}TE <TY SET,TY T> IN bool In(CO SET& S,CO T& t){RE S.count(t)== 1;}DF_OF_POP_FOR_SET(set<T>);DF_OF_POP_FOR_SET(unordered_set<T>);DF_OF_POP_FOR_SET(multiset<T>);DF_OF_POP_FOR_SET(unordered_multiset<T>);DF_OF_UNION_FOR_SET(set<T>);DF_OF_UNION_FOR_SET(unordered_set<T>);DF_OF_UNION_FOR_SET(multiset<T>);DF_OF_UNION_FOR_SET(unordered_multiset<T>);DF_OF_UNION_FOR_SET(VE<T>);DF_OF_UNION_FOR_SET(LI<T>);
#endif

/* Tuple (5KB)*/
#define DF_OF_AR_FOR_TUPLE(OPR)TE <TY T,TY U,TE <TY...> TY V> IN auto OP OPR ## =(V<T,U>& t0,CO V<T,U>& t1)-> decltype((get<0>(t0),t0))&{get<0>(t0)OPR ## = get<0>(t1);get<1>(t0)OPR ## = get<1>(t1);RE t0;}TE <TY T,TY U,TY V> IN tuple<T,U,V>& OP OPR ## =(tuple<T,U,V>& t0,CO tuple<T,U,V>& t1){get<0>(t0)OPR ## = get<0>(t1);get<1>(t0)OPR ## = get<1>(t1);get<2>(t0)OPR ## = get<2>(t1);RE t0;}TE <TY T,TY U,TY V,TY W> IN tuple<T,U,V,W>& OP OPR ## =(tuple<T,U,V,W>& t0,CO tuple<T,U,V,W>& t1){get<0>(t0)OPR ## = get<0>(t1);get<1>(t0)OPR ## = get<1>(t1);get<2>(t0)OPR ## = get<2>(t1);get<3>(t0)OPR ## = get<3>(t1);RE t0;}TE <TY ARG,TY T,TY U,TE <TY...> TY V> IN auto OP OPR ## =(V<T,U>& t0,CO ARG& t1)-> decltype((get<0>(t0),t0))&{get<0>(t0)OPR ## = t1;get<1>(t0)OPR ## = t1;RE t0;}TE <TY ARG,TY T,TY U,TY V> IN tuple<T,U,V>& OP OPR ## =(tuple<T,U,V>& t0,CO ARG& t1){get<0>(t0)OPR ## = t1;get<1>(t0)OPR ## = t1;get<2>(t0)OPR ## = t1;RE t0;}TE <TY ARG,TY T,TY U,TY V,TY W> IN tuple<T,U,V,W>& OP OPR ## =(tuple<T,U,V,W>& t0,CO ARG& t1){get<0>(t0)OPR ## = t1;get<1>(t0)OPR ## = t1;get<2>(t0)OPR ## = t1;get<3>(t0)OPR ## = t1;RE t0;}TE <TE <TY...> TY V,TY...ARGS,TY ARG> IN auto OP OPR(CO V<ARGS...>& t0,CO ARG& t1)-> decldecay_t((get<0>(t0),t0)){auto t = t0;RE MO(t OPR ## = t1);}
#define DF_OF_INCREMENT_FOR_TUPLE(INCR)TE <TY T,TY U,TE <TY...> TY V> IN auto OP INCR(V<T,U>& t)-> decltype((get<0>(t),t))&{INCR get<0>(t);INCR get<1>(t);RE t;}TE <TY T,TY U,TY V> IN tuple<T,U,V>& OP INCR(tuple<T,U,V>& t){INCR get<0>(t);INCR get<1>(t);INCR get<2>(t);RE t;}TE <TY T,TY U,TY V,TY W> IN tuple<T,U,V,W>& OP INCR(tuple<T,U,V,W>& t){INCR get<0>(t);INCR get<1>(t);INCR get<2>(t);INCR get<3>(t);RE t;}
TE <CL Traits,TY T> IN IS& OP>>(IS& is,tuple<T>& arg){RE is >> get<0>(arg);}TE <CL Traits,TY T,TY U,TE <TY...> TY V> IN auto OP>>(IS& is,V<T,U>& arg)-> decltype((get<0>(arg),is))&{RE is >> get<0>(arg)>> get<1>(arg);}TE <CL Traits,TY T,TY U,TY V> IN IS& OP>>(IS& is,tuple<T,U,V>& arg){RE is >> get<0>(arg)>> get<1>(arg)>> get<2>(arg);}TE <CL Traits,TY T,TY U,TY V,TY W> IN IS& OP>>(IS& is,tuple<T,U,V,W>& arg){RE is >> get<0>(arg)>> get<1>(arg)>> get<2>(arg)>> get<3>(arg);}TE <CL Traits,TY T> IN OS& OP<<(OS& os,CO tuple<T>& arg){RE os << get<0>(arg);}TE <CL Traits,TY T,TY U,TE <TY...> TY V> IN auto OP<<(OS& os,CO V<T,U>& arg)-> decltype((get<0>(arg),os))&{RE os << get<0>(arg)<< " " << get<1>(arg);}TE <CL Traits,TY T,TY U,TY V> IN OS& OP<<(OS& os,CO tuple<T,U,V>& arg){RE os << get<0>(arg)<< " " << get<1>(arg)<< " " << get<2>(arg);}TE <CL Traits,TY T,TY U,TY V,TY W> IN OS& OP<<(OS& os,CO tuple<T,U,V,W>& arg){RE os << get<0>(arg)<< " " << get<1>(arg)<< " " << get<2>(arg)<< " " << get<3>(arg);}DF_OF_AR_FOR_TUPLE(+);TE <TY T,TY U,TE <TY...> TY V> IN auto OP-(CO V<T,U>& t)-> decltype(get<0>(t),t){RE{-get<0>(t),-get<1>(t)};}TE <TY T,TY U,TY V> IN tuple<T,U,V> OP-(CO tuple<T,U,V>& t){RE{-get<0>(t),-get<1>(t),-get<2>(t)};}TE <TY T,TY U,TY V,TY W> IN tuple<T,U,V,W> OP-(CO tuple<T,U,V,W>& t){RE{-get<0>(t),-get<1>(t),-get<2>(t),-get<3>(t)};}DF_OF_AR_FOR_TUPLE(-);DF_OF_AR_FOR_TUPLE(*);DF_OF_AR_FOR_TUPLE(/);DF_OF_AR_FOR_TUPLE(%);DF_OF_INCREMENT_FOR_TUPLE(++);DF_OF_INCREMENT_FOR_TUPLE(--);

#define DF_OF_HASH_FOR_TUPLE(PAIR)TE <TY T,TY U> IN size_t hash<PAIR<T,U>>::OP()(CO PAIR<T,U>& n)CO{ST CO size_t seed =(GetRand(1e3,1e8)<< 1)| 1;ST CO hash<T> h0;ST CO hash<U> h1;RE(h0(get<0>(n))* seed)^ h1(get<1>(n));}
TE <TY T> DC_OF_HASH(tuple<T>);TE <TY T,TY U> DC_OF_HASH(pair<T,U>);TE <TY T,TY U> DC_OF_HASH(tuple<T,U>);TE <TY T,TY U,TY V> DC_OF_HASH(tuple<T,U,V>);TE <TY T,TY U,TY V,TY W> DC_OF_HASH(tuple<T,U,V,W>);
TE <TY T> IN size_t hash<tuple<T>>::OP()(CO tuple<T>& n)CO{ST CO hash<T> h;RE h(get<0>(n));}DF_OF_HASH_FOR_TUPLE(pair);DF_OF_HASH_FOR_TUPLE(tuple);TE <TY T,TY U,TY V> IN size_t hash<tuple<T,U,V>>::OP()(CO tuple<T,U,V>& n)CO{ST CO size_t seed =(GetRand(1e3,1e8)<< 1)| 1;ST CO hash<pair<T,U>> h01;ST CO hash<V> h2;RE(h01({get<0>(n),get<1>(n)})* seed)^ h2(get<2>(n));}TE <TY T,TY U,TY V,TY W> IN size_t hash<tuple<T,U,V,W>>::OP()(CO tuple<T,U,V,W>& n)CO{ST CO size_t seed =(GetRand(1e3,1e8)<< 1)| 1;ST CO hash<pair<T,U>> h01;ST CO hash<pair<V,W>> h23;RE(h01({get<0>(n),get<1>(n)})* seed)^ h23({get<2>(n),get<3>(n)});}

/* Vector (3KB)*/
#define DC_OF_COUT_FOR_VE(V)TE <CL Traits,TY Arg> IN OS& OP<<(OS& os,CO V<Arg>& arg)
#define DF_OF_COUT_FOR_VE(V)TE <CL Traits,TY Arg> IN OS& OP<<(OS& os,CO V<Arg>& arg){auto BE = arg.BE(),EN = arg.EN();auto IT = BE;WH(IT != EN){(IT == BE?os:os << " ")<< *IT;IT++;}RE os;}
DF_OF_COUT_FOR_VE(VE);DF_OF_COUT_FOR_VE(LI);DF_OF_COUT_FOR_VE(set);DF_OF_COUT_FOR_VE(unordered_set);DF_OF_COUT_FOR_VE(multiset);IN VO VariadicResize(CRI SZ){}TE <TY Arg,TY... ARGS> IN VO VariadicResize(CRI SZ,Arg& arg,ARGS&... args){arg.resize(SZ);VariadicResize(SZ,args...);}

#define DF_OF_SCALAR_ACTION_FOR_VE(V,OPR)TE <TY T> IN V<T>& OP OPR ## =(V<T>& a,CO T& t){for(auto& s:a){a OPR ## = t;}RE a;}
#define DF_OF_AR_FOR_VE(V,OPR)TE <TY T> IN V<T>& OP OPR ## =(V<T>& a0,CO V<T>& a1){AS(a0.SZ()<= a1.SZ());auto IT0 = a0.BE(),EN0 = a0.EN();auto IT1 = a1.BE();WH(IT0 != EN0){*(IT0++)OPR ## = *(IT1++);}RE a0;}TE <TY T,TY U> IN V<T> OP OPR(V<T> a,CO U& u){RE MO(a OPR ## = u);}
#define DF_OF_INCREMENT_FOR_VE(V,INCR)TE <TY T> IN V<T>& OP INCR(V<T>& a){for(auto& i:a){INCR i;}RE a;}
#define DF_OF_ARS_FOR_VE(V)TE <TY T> IN V<T>& OP+=(V<T>& a,CO T& t){a.push_back(t);RE a;}DF_OF_SCALAR_ACTION_FOR_VE(V,*);DF_OF_SCALAR_ACTION_FOR_VE(V,/);DF_OF_SCALAR_ACTION_FOR_VE(V,%);DF_OF_AR_FOR_VE(V,+);DF_OF_AR_FOR_VE(V,-);DF_OF_AR_FOR_VE(V,*);DF_OF_AR_FOR_VE(V,/);DF_OF_AR_FOR_VE(V,%);DF_OF_INCREMENT_FOR_VE(V,++);DF_OF_INCREMENT_FOR_VE(V,--);TE <TY T> IN V<T> OP*(CO T& scalar,V<T> v){for(auto& t:v){t *= scalar;}RE MO(v);}
DF_OF_ARS_FOR_VE(VE);DF_OF_ARS_FOR_VE(LI);TE <TY V> IN auto Get(V& a){RE[&](CRI i = 0)-> CO decldecay_t(a[0])&{RE a[i];};}TE <TY T> IN VE<T> id(CRI SZ){VE<T> AN(SZ);for(int i = 0;i < SZ;i++){AN[i]= i;}RE AN;}TE <TY T> IN VO Sort(VE<T>& a,CO bool& reversed = false){if(reversed){ST auto comp =[](CO T& t0,CO T& t1){RE t1 < t0;};sort(a.BE(),a.EN(),comp);}else{sort(a.BE(),a.EN());}}TE <TY T0,TY T1> IN VO Sort(VE<T0>& a,VE<T1>& b,CO bool& reversed = false){CO int SZ = a.SZ();AS(SZ == int(b.SZ()));VE<pair<T0,T1>> v(SZ);for(int i = 0;i < SZ;i++){v[i]={MO(a[i]),MO(b[i])};}Sort(v,reversed);for(int i = 0;i < SZ;i++){a[i]= MO(v[i].first);b[i]= MO(v[i].second);}}TE <TY T> IN VE<int> IndexSort(CO VE<T>& a,CO bool& reversed = false){auto index = id<int>(a.SZ());if(reversed){sort(index.BE(),index.EN(),[&](CRI i,CRI j){RE a[j]< a[i];});}else{sort(index.BE(),index.EN(),[&](CRI i,CRI j){RE a[i]< a[j];});}RE index;}

/* Map (1KB)*/
#define DF_OF_AR_FOR_MAP(MAP,OPR)TE <TY T,TY U> IN MAP<T,U>& OP OPR ## =(MAP<T,U>& a,CO pair<T,U>& v){a[v.first]OPR ## = v.second;RE a;}TE <TY T,TY U> IN MAP<T,U>& OP OPR ## =(MAP<T,U>& a0,CO MAP<T,U>& a1){for(auto&[t,u]:a1){a0[t]OPR ## = u;}RE a0;}TE <TY T,TY U,TY ARG> IN MAP<T,U> OP OPR(MAP<T,U> a,CO ARG& arg){RE MO(a OPR ## = arg);}
#define DF_OF_ARS_FOR_MAP(MAP)DF_OF_AR_FOR_MAP(MAP,+);DF_OF_AR_FOR_MAP(MAP,-);DF_OF_AR_FOR_MAP(MAP,*);DF_OF_AR_FOR_MAP(MAP,/);DF_OF_AR_FOR_MAP(MAP,%);
TE <TY T,TY U>US Map = conditional_t<is_COructible_v<unordered_map<T,int>>,unordered_map<T,U>,conditional_t<is_ordered::value<T>,map<T,U>,VO>>;
DF_OF_ARS_FOR_MAP(map);DF_OF_ARS_FOR_MAP(unordered_map);

/* StdStream (2KB)*/
TE <CL Traits> IN IS& VariadicCin(IS& is){RE is;}TE <CL Traits,TY Arg,TY... ARGS> IN IS& VariadicCin(IS& is,Arg& arg,ARGS&... args){RE VariadicCin(is >> arg,args...);}TE <CL Traits> IN IS& VariadicSet(IS& is,CRI i){RE is;}TE <CL Traits,TY Arg,TY... ARGS> IN IS& VariadicSet(IS& is,CRI i,Arg& arg,ARGS&... args){RE VariadicSet(is >> arg[i],i,args...);}TE <CL Traits> IN IS& VariadicGetline(IS& is,CO char& separator){RE is;}TE <CL Traits,TY Arg,TY... ARGS> IN IS& VariadicGetline(IS& is,CO char& separator,Arg& arg,ARGS&... args){RE VariadicGetline(getline(is,arg,separator),separator,args...);}TE <CL Traits,TY Arg> IN OS& VariadicCout(OS& os,Arg&& arg){RE os << forward<Arg>(arg);}TE <CL Traits,TY Arg1,TY Arg2,TY... ARGS> IN OS& VariadicCout(OS& os,Arg1&& arg1,Arg2&& arg2,ARGS&&... args){RE VariadicCout(os << forward<Arg1>(arg1)<< " ",forward<Arg2>(arg2),forward<ARGS>(args)...);}TE <CL Traits,TY Arg> IN OS& VariadicCoutNonSep(OS& os,Arg&& arg){RE os << forward<Arg>(arg);}TE <CL Traits,TY Arg1,TY Arg2,TY... ARGS> IN OS& VariadicCoutNonSep(OS& os,Arg1&& arg1,Arg2&& arg2,ARGS&&... args){RE VariadicCoutNonSep(os << forward<Arg1>(arg1),forward<Arg2>(arg2),forward<ARGS>(args)...);}TE <CL Traits,TY ARRAY> IN OS& CoutArray(OS& os,CRI i_start,CRI i_ulim,ARRAY&& a){for(int i = i_start;i < i_ulim;i++){(i == i_start?os:(os << " "))<< a[i];}RE os;}

/* ConstexprModulo (7KB)*/
#define RP Represent
#define DeRP Derepresent

/* Sum (2KB) */
TE <TY T,TE <TY...> TY V,TY OPR> T LeftConnectiveProd(CO V<T>& f,OPR opr){AS(!f.empty());auto IT = f.BE(),EN = f.EN();T AN = *(IT++);WH(IT != EN){AN = opr(MO(AN),*(IT++));}RE AN;}TE <TY T,TE <TY...> TY V> IN T Sum(CO V<T>& f){RE LeftConnectiveProd(f,[](T t0,CO T& t1){RE MO(t0 += t1);});}TE <TY T,TE <TY...> TY V> IN T Prod(CO V<T>& f){RE LeftConnectiveProd(f,[](T t0,CO T& t1){RE MO(t0 *= t1);});}TE <TY T,TE <TY...> TY V> IN T Max(CO V<T>& f){RE *max_element(f.BE(),f.EN());}TE <TY T,TE <TY...> TY V> IN T Min(CO V<T>& f){RE *min_element(f.BE(),f.EN());}TE <TY T,TY UINT>T Power(T t,UINT EX,T init = 1){(EX & 1)== 1?init *= t:init;EX >>= 1;WH(EX > 0){t = Square(t);(EX & 1)== 1?init *= t:init;EX >>= 1;}RE MO(init);}TE <TY INT> IN INT ArithmeticProgressionSum(CO INT& l,INT r,CO INT& d = 1){AS(l <= r);CO INT c =(r - l)/ d;RE(c & 1)== 0?(c + 1)*(l + d *(c >> 1)):((c + 1)>> 1)*((l << 1)+ d * c);}TE <TY INT> IN INT ArithmeticProgressionSum(CO INT& r){RE ArithmeticProgressionSum(INT{},r);}TE <TY T,TY UINT> IN T GeometricProgressionSum(T rate,UINT EX_max,CO T& init = 1){T rate_minus = rate - 1;RE rate_minus == 0?init * ++EX_max:(Power(MO(rate),MO(++EX_max))- 1)/ MO(rate_minus)* init;}TE <TY T,TY UINT>T GeometricProgressionLinearCombinationSum(VE<T> rate,VE<UINT> EX_max,CO VE<T>& init){CO int SZ = init.SZ();AS(int(rate.SZ())== SZ && int(EX_max.SZ())== SZ);T AN{};for(int i = 0;i < SZ;i++){AN += GeometricProgressionSum(MO(rate[i]),MO(EX_max[i]),init[i]);}RE AN;}
#endif
/* AAA 常設ライブラリは以上に挿入する。*/

#define INCLUDE_LIBRARY
#include __FILE__
#endif /* INCLUDE_LIBRARY */
#endif /* INCLUDE_SUB */
#endif /* INCLUDE_MAIN */